I have, however, seen restaurant lobby groups threatening an unlikely robot takeover of restaurant labor if minimum wages should rise:
https://mgtvkron.files.wordpress.com/2014/07/bad-idea-billbo...
I suspect progress in automation follows it's own path and when it destroys jobs, it destroys jobs such that no amount of wage cutting will impede its progress.
Labor efficiency is not just about having a robot take your order, it's the difference between a McDonald's where all the parts are pre-assembles elsewhere and a mom and pop joint where someone is hand-forming the patties.
Anyway, there is another article in this thread that explains how minimum wage increases tend to force the less efficient mom and pop joint out of business, thus accelerating the labor efficiency progress.
I'm for minimum wage increases, but I also believe that certain jobs will not exist with a high enough minimum wage. Any increase in minimum wage probably needs to be met with an increase in training/education.
In a competitive market where using less labor is not an option, wouldn't #2 happen? All businesses would have the same cost increase and could expect to raise prices.